Method for operating a digital mobile radio network with space-time block codes

ABSTRACT

The invention relates to the preparation and numerical optimization of non-linear space-time block codes for application in a digital mobile radio system, with a maximum transmission diversity for “Rate 1” transmission systems in the case of two or more transmitter antennae (Txn) and with complex symbols.

[0001] The present invention relates to a method for operating a digital mobile radio network using space-time block codes (ST codes) according to the preamble of Claim 1.

[0002] Such methods according to the preamble of Claim 1 are for example disclosed in “Space-Time Codes for High Data Rate Wireless Communication: Performance Criterion and Code Construction” by V. Tarokh et al. in IEEE Transactions on Information Theory, vol. 44, no. 2 March 1998, pages 744-765 or in “Space-Time Block Codes from Orthogonal Designs”, IEEE Transactions on Information Theory, vol. 45, no. 5, July 1999, pages 1456-1467 by V. Tarokh et al.

[0003] Use of the time-space block codes disclosed there is subject to the following physical problem in mobile radio networks: the high level of attenuation or distortion caused by the transmission path of a transmission signal transmitted in a wireless manner from a transmitting station (e.g. a fixed base station in a mobile radio network) makes it very difficult for a receiving station (e.g. a mobile station in a mobile radio network) to identify the originally transmitted transmission signal correctly. This applies in particular, when the receiving mobile station is located in a multipath wireless environment, where it only receives a number of highly attenuated echoes of the original transmission signal due to multiple reflections of the original transmission signal off the walls of surrounding buildings. To rectify this, a certain “diversity” of the received signal has to be ensured. This is achieved by the receiving station also being provided with a further one or more less highly attenuated “images” of the transmitted transmission signal. What is known as the “diversity order” here is a measure of the number of statistically independent “images” of the transmission signals transmitted by the transmitter received at the receiver.

[0004] In practice this necessary diversity is achieved for example by using a number of transmitter or receiver antennas each positioned spatially apart from each other or with different polarities on the transmitter and/or receiver side, each of which transmits or receives an “image” of a signal to be transmitted. “Image” here does not necessarily mean that two or more exactly identical copies of the same signal are transmitted.

[0005] Instead, even with the known space-time block codes, different signals can be transmitted at a defined time from the different transmitter antennas, each of said signals being created by a specific algorithm from the data bits of the original signal. Each receiver antenna then receives the sum of the transmission signals, modified due to the transmission path, at a defined (later due to the signal runtime) time, from which signals those data bits, which correspond with the greatest likelihood to the data bits of the original signal, are reconstructed on the receiver side using MLD (maximum likelihood detection) estimate algorithms.

[0006] See FIGS. 1 and 2 for a clearer explanation of the problem underlying the present invention.

[0007]FIG. 1 shows a diagrammatic illustration of a conventional single antenna to single antenna radio transmission link between a transmitter Tx and a receiver Rx in a digital mobile radio network.

[0008] Data bits are fed into the transmitter, comprising for example a series of ones and zeros. Let it be assumed that a bit vector (1,0,1) of length l=3 is fed in, comprising the series 1-0-1. Depending on the coding method used, this bit vector is transformed on the transmitter side with one-to-one correspondence (i.e. with reversible uniqueness) into a symbol vector of length n. As shown in FIG. 1 by different shades of gray, the figures occurring in the symbol vector do not have to be 1 and 0, in particular the figures occurring in the symbol vector can also be complex figures, with which for example the signals corresponding to the real component and the imaginary component are sent with a phase offset of 90° in respect of each other.

[0009]FIG. 1 shows the case where n=l=3, i.e. each individual bit is coded into an individual symbol (BPSK modulation—binary phase shift keying; with QPSK—quadrature phase shift keying n=l/2 would apply).

[0010] A symbol vector comprising a number of individual symbols is then transmitted by an antenna in the transmitter Tx and received by a receiver Rx. The symbol vector of length n=3 is reconstructed there by a reverse transformation into the original bit vector (here (1,0,1)) of length l=3.

[0011]FIG. 2 shows a diagrammatic illustration of a radio transmission link between a transmitter unit with n=3 transmitter antennas Tx1, Tx2, Tx3 and a receiver Rx in a digital mobile radio network, with which space-time block codes already known per se are used. A bit vector of length l=3 is fed to a space-time block coding device (ST coder=space time coder). This maps an incoming bit vector onto an n×n matrix.

[0012] Here n corresponds to the number of transmitter antennas used in the transmitter unit. In a defined time slot j an antenna i sends a signal, which corresponds to the matrix element c_(ijk) of a 3×3 matrix C_(k), which was coded by the ST coder from the incoming vector of length l=3. k here is an index, which differentiates individual matrices, which in turn correspond for their part to k different bit vectors. These relationships are described in more detail below.

[0013] As a result of the signals transmitted via the three transmitter antennas Tx1, Tx2 and Tx3, signals arrive at the receiver antenna Rx at a receiving time corresponding to the transmission time slot j (later due to runtime), said signals corresponding to the matrix elements c_(1jk), c_(2jk), c_(3jk). On the receiver side a space-time block matrix C_(k) is reconstructed from the sum of the incoming signals in a space-time block decoding device (ST decoder) by means of MLD algorithms and translated back by reverse mapping into the corresponding original bit vector (here (1,0,1)).

[0014] The general problem here is maximizing diversity at the mobile station of a digital mobile radio network by using a number of transmitter antennas at the base station. No specific prior knowledge about the downlink channel (from a base station to a mobile station), which changes over time, should be assumed.

[0015] In the case of linear space-time block codes with two antennas, the result is known (it is optimal in the sense that it doubles the diversity for two antennas) and it has become part of the UMTS standard (3GPP TS 25.211 V3.4.0: Physical channels and mapping of transport channels onto physical channels (FDD) (1999 edition), September 2000). This known space-time block code scheme satisfies the “rate 1” requirement.

[0016] A “rate 1” space-time block code scheme can be described as a system, in which, for each time interval considered, precisely the same number of data bits can be sent through effectively from a transmitter to a receiver as with the reference system shown in FIG. 1 with only one transmitter and one receiver antenna. In other words, a “rate 1” ST code has the same transmission rate compared with a basic system with only one transmitter and one receiver antenna. This is for example a system in which a block of two code words is transmitted to the receiver at the same time in two successive time windows via two different transmission channels; the receiver thereby receives precisely the same amount of information per unit of time as if two corresponding individual bits had been transmitted in two successive time windows via a single transmission channel. Provision is therefore made in the W-CDMA mode of the UMTS standard for future mobile radio systems (see for example www.3GPP.org) to transmit the standard mapping of four bits onto 2⁴=16 ST symbols in two time stages via two transmitter antennas. A “rate 1” system for external blocks therefore behaves in exactly the same way as the basic system with only one transmitter and one receiver antenna. This characteristic is decisive when upgrading a basic system to a system with a number of transmitter antennas and space-time block codes.

[0017] The space-time block code used in W-CDMA mode corresponds to what is known as the Alamouti code. This is a code that is very simple to reconstruct on the receiver side to increase diversity in a digital mobile radio network with a transmitting station with two (n=2) transmitter antennas. The Alamouti code is for example disclosed in “A simple Transmitter Diversity Scheme for Wireless Communications”, IEEE J. Select Areas Commun, vol. 16, pages 1451-1458, October 1998 by S. M. Alamouti or in the two publications by V. Tarokh referred to above.

[0018] In order to increase diversity in a mobile radio network further in the future both in the uplink direction (from,a mobile station to a base station) and in the downlink direction (from a base station to a mobile station), the number of antennas per sector of a base station should be greater than two. It is therefore clear that space-time codes are required, which can be used in the case of three, four or more transmitter antennas. An increase in diversity with the same transmission power results in an increase in receiving quality. Or looked at another way, an increase in diversity with the same receiving quality means a reduction in transmission power. The transmission power then not used up in a transmitter can in turn be used to supply more users.

[0019] The performance of a space-time block code is also influenced by the intervals between the code words. Observations on this are contained in the two articles by V. Tarokh referred to above.

[0020] In the transmission shown diagrammatically in FIG. 2 the matrix shown with the elements c_(ijk) corresponds to a code word with the number k. The “interval” between code words, i.e. the “interval” between two matrices C_(k1) and C_(k2) is ultimately a measure of the quality of a transmission code, i.e. the likelihood with which the originally transmitted code words can be reconstructed as uniquely as possible from a sequence of code words received at the receiver with distortion due to the transmission path. In this respect the known space-time blocks in the case of two transmitter antennas can be further improved.

[0021] In the case of three, four or more antennas (n>2) it is shown in “Space-time Block Codes from Orthogonal Designs”, IEEE Transactions on Information Theory, vol. 45, no. 5, July 1999, pages 1456-1467 by V. Tarokh et al. that linear codes with “rate 1” and complex symbols cannot exist.

[0022] In a presentation given at Globecom 2000 in December 2000 on “Complex Space-Time Block Codes for Four Tx Antennas” by Olav Tirkkonen and Ari Hottinen, the case of n=4 transmitter antennas was examined and complex space-time block codes specified with a rate of 3/4.

[0023] The search for space-time codes, in particular for a number n>2 of transmitter antennas, has therefore taken two directions:

[0024] 1. In “Space-time Block Codes from Orthogonal Designs”, IEEE Transactions on Information Theory, vol. 45, no. 5, July 1999, pages 1456-1467 by V. Tarokh et al., linear space-time codes with a rate less than 1 are constructed for more than two transmitter antennas. Here the space-time symbols are linear combinations of the original signals. These constructions are not available for external coding, because they do not have the “rate 1” characteristic. This means that they cannot be integrated as simple additional features in an existing mobile radio system without space-time coding.

[0025] 2. Space-time trellis codes are constructed by combining space-time codes with external error correction codes. Combining space-time mapping with external coding however makes it impossible to integrate space-time codes as a simple additional feature in an existing system. Also in such a case proven external coding techniques such as turbocodes (see C. Berron, A. Glavieux and P. Thitimajshima: Near Shannon limit error correcting codes and decoding: turbo codes” in Proc. IEEE ICC, Geneva, May 1993, pages 1064-1070)) or trellis codes must be modified.

[0026] This reveals the problem that the digital code words to be transmitted between the transmitting and receiving stations in a digital mobile radio network should be optimized in the case of two or more transmitter antennas with regard to the following:

[0027] 1. There should be a “rate 1” code if possible.

[0028] This is a mandatory requirement in order to be able to upgrade mobile radio networks already commercially available with the lowest possible upgrade costs for the use of the new space-time block codes. When “rate 1” is used, when upgrading from a conventional single antenna to single antenna system shown in FIG. 1 to a multi-antenna to single antenna system as shown in FIG. 2 (or even to multi-antenna to multi-antenna systems), the assemblies (not shown in FIGS. 1 and 2) required on the transmitter side to generate the bit vector to be fed in and the assemblies (also not shown) required on the receiver side to further process the reconstructed bit vectors output by the receiver unit do not have to be changed. “Rate 1” codes thus guarantee “downward compatibility” of digital mobile radio stations operated with multi-antenna units and corresponding space-time block codes with other already existing system components, thereby eliminating a significant obstacle to investment impeding the practical introduction of “rate≠1” systems, as with these the assemblies for transmitter-side bit vector generation and for receiver-side bit vector reconstruction must also be changed.

[0029] The introduction of “rate 1” space-time blocks means that it is also possible to leave the other parameters of a transmitter code scheme (such as channel coding, interleaving, service multiplexing, etc. unchanged).

[0030] 2. The code should be simple to construct on the transmitter side and simple to reconstruct on the receiver side.

[0031] 3. The code words should have the “biggest interval” possible between them. This means that a set of code words should be structured so that from the signals received on the receiver side with noise and/or distortion, each of which comprises the original signal transmitted on the transmitter side times a “fading factor” describing the fading of intensity associated with increasing distance plus noise (thermal noise at the input amplifier of the receiver plus interference noise due to disruptive signals from other users of the mobile radio network), [lacuna] can be reconstructed even with a relatively high level of interference in the most error-free possible manner as the signals transmitted on the transmitter side (i.e. without confusion between individual code words).

[0032] 4. The space-time block code used should maximize diversity, i.e. for two transmitter antennas the theoretically maximum degree of diversity 2 should be achieved if possible, for three transmitter antennas the theoretically maximum degree of diversity 3, etc.

[0033] 5. The space-time block code used should allow complex transmission symbols, in order for example that it can be used with UMTS where QPSK (quadrature phase shift keying) modulation is used. Complex symbols also allow 8-PSK (8-phase shift keying) or M-QAM (M-fold quadrature amplitude modulation).

[0034] The object of the present invention is to optimize the known methods for operating a digital mobile radio network according to the preamble of Claim 1 as far as possible by using alternative space-time block codes in respect of the above criteria 1 to 5. In particular “rate 1” space-time block codes with maximum transmission diversity should be prepared in the case of three or more transmitter antennas (n>2). Space-time block codes with interval optimization should also be prepared in particular in the case n≧2.

[0035] This object is achieved in its most general form according to the invention by the measures in Claim 1.

[0036] The measures in Claim 1 are based on the knowledge presented in the further disclosure that in the case of known methods for operating mobile radio networks according to the preamble of Claim 1 space-time block codes can be created, which can be constructed from unitary n×n matrices according to the characterizing features of Claim 1.

[0037] In particular in the case of n>2, i.e. three and more transmitter antennas, it is demonstrated according to the present invention that space-time block codes for mobile radio systems with n transmitter antennas and m receiver antennas with a complex ST modulation scheme and with the degree of diversity n×m and “rate 1” essentially exist, as they can be constructed from unitary n×n matrices according to the characterizing section of Claim 1.

[0038] The further independent method Claims 3 and 8 relate to methods according to the preamble of Claim 1, with which in the case of two and more (n≧2) transmitter antennas optimized space-time block codes with a complex ST modulation scheme with the degree of diversity n×m and “rate 1” are determined numerically, with which the code symbols have the greatest possible “interval” in the sense of a metric specified in each instance.

[0039] The further independent Claims pertaining to the matter relate to base and mobile stations in a digital mobile radio network, in which reference tables, which contain the matrix elements of code words used in the methods according to the invention, are stored as well as computer program products in which corresponding reference tables are implemented.

[0040] The dependent claims relate to advantageous embodiments of the present invention.

[0041] According to the present invention measures are specified in particular for the construction of “rate 1” space-time block codes for three, four or more antennas, with which a maximum diversity amplification is achieved.

[0042] This is considered to be a significant theoretical and practical breakthrough and is based on two changes compared with the procedures known from the prior art:

[0043] 1. Non-linear codes are used. This does not represent a problem, as the number of code words in space-time blocks is 2″ for BPSK (binary phase shift keying); digital frequency modulation technique for sending data via a coaxial cable network: this type of modulation is less efficient but also less subject to noise than similar modulation techniques, such as for example QPSK (quadrature phase shift keying) and 4″ for QPSK modulation. In other words even for n=4, when a QPSK modulation is used, only 256 code words are used and these can be stored easily in a table.

[0044] 2. Neither the transmitted energy per antenna and time unit nor the energy conveyed via all the antennas at a predetermined time is kept constant. This is not critical with W-CDMA in particular, as greater fluctuations can also occur in the emitted energy in a standard system due to the superimposition of different user signals. The additional fluctuations in power, introduced by the use of new space-time codes, are negligible in comparison.

DESCRIPTION OF FIGURES

[0045]FIG. 1 shows a diagrammatic illustration of a radio transmission link and a corresponding symbol vector coding and decoding for a known single antenna to single antenna arrangement in a digital mobile radio network;

[0046]FIG. 2 shows a diagrammatic illustration of a radio transmission link and a corresponding symbol coding and decoding for a space-time block coding known per se in a multi-antenna to single antenna arrangement in a digital mobile radio network;

[0047]FIG. 3 shows in table form a specific example of a BPSK-compliant coding of eight bit vectors into the matrix elements of eight complex unitary matrices (ST symbols), which are used as the basis for a space-time block coding during the implementation of a method according to the invention for n=3 transmitter antennas;

[0048]FIG. 4 shows the intervals between eight ST symbols in the case of a space-time block coding discussed in relation to FIG. 3.

[0049]FIG. 5 shows the comparison in the code spectrum for an inventively constructed interval-optimized space-time block code for the case SU(2) in the case of n=2 transmitter antennas and a space-time code constructed according to the known Alamouti scheme, which is not interval-optimized;

[0050]FIG. 6 shows a code spectrum for an inventively constructed interval-optimized space-time code for the case SU(3) in the case of n=3 transmitter antennas and QPSK modulation using a minimum standard;

[0051]FIG. 7 shows a code spectrum for space-time block codes constructed by means of inventive numerical optimization methods in the case of n=3 transmitter antennas and BPSK modulation using different standards;

[0052]FIG. 8 shows a BPSK simulation for two and three antennas using an L_(min) code;

[0053]FIG. 9 shows the spectrum of minimum eigenvalues with different inventive codes for n=3 transmitter antennas and BPSK modulation; and

[0054]FIG. 10 shows a BPSK simulation for n=3 transmitter antennas using an L⁻¹ code.

[0055] To give a better understanding of the present invention, constructive proof is first provided to show that complex space-time block codes with maximum diversity exist not only for the case n=2 transmitter antennas (as shown for example by the known Alamouti scheme) but that complex space-time block codes with maximum diversity also exist for the case with n>2 transmitter antennas.

[0056] The mathematical structure on which the present invention is based is relatively complicated. In practice application of the present invention is however reduced to the use of a reference table for space-time code words, which are stored in the receiver and transmitter. FIG. 3 shows an example of such a reference table. Its structure and method of use are explained in more detail below. These tables can then be used without a deeper knowledge of their without a deeper knowledge of their derivation being required.

[0057] The mathematical notation used in the present application is for example explained in Simon Barry, “Representation of Finite and Compact Groups”, Graduate Studies in Mathematics, Volume 10, American Mathematical Society or in Bronstein, Semendjajew: “Taschenbuch der Mathematik” (Pocket Book of Mathematics), p. 155. published by Harri Deutsch, Thun and Frankfurt/Main, ISBN 3 87144 492 8.

[0058] According to the invention a conceptually new approach is defined for space-time codes. First the transmitter units are disclosed. Next a standard multi-input-multi-output fading channel is disclosed. Finally the process for recording space-time symbols is disclosed. On this basis constructive proof of the existence of space-time codes with maximum diversity is given for mobile radio systems with n≧2 transmitter antennas. This proof covers cases with complex and real space-time symbols.

[0059] Although the proof of existence is constructive, it does not provide an optimum space-time code, i.e. a space-time code with optimized “intervals” between the code words. Therefore practical methods for constructing codes based on numerical optimization are defined. Finally the results are also confirmed by simulations.

Transmitter

[0060] Let us assume first that there are n transmitter antennas and one receiver antenna.

[0061] Extension to n transmitter antennas and m receiver antennas looks like this:

[0062] In the case of m>1 receiver antennas, the codes and detection methods disclosed below can be implemented separately in each instance for all the m receiver antennas. The results are then combined in what is known as a “maximum ratio combining” method. Such a “maximum ratio combining” method is for example described in J. Proakis, M. Salehi: Communication Systems Engineering, Prentice Hall, 1994, ISBN: 0-13-306625-5.

[0063] This then gives a maximum degree of diversity of n×m. In particular the extension to m>1 receiver antennas does not influence optimization of the transmitter side. This can be identified for example from the equation (8) in Tarokh et al., “Space-Time Codes for high data wireless communication: performance criterion and code construction”. Here it is shown that the likelihood of mutilation between two sequences is equal to the product of the likelihood of mutilation for only one receiver antenna in each instance and all product components are of equal size.

[0064] Correspondingly the minimization of a product component results in the minimization of the product as a whole. This means that the number of receiver antennas has no influence on the optimality of the transmission symbols.

[0065] To this extent all the transmission methods disclosed below can be used without change for any number of receiver antennas.)

[0066] A space-time block coding (“block ST modulation”) is described as the mapping of a total of 2^(l) different bit vectors {right arrow over (b)}∈B^(l) (each of which comprise 1 bits ∈{0,1}) onto a set of 2^(l) space-time symbols C({right arrow over (q)})∈U(n), which are described as unitary n×n matrices by the mapping:

STM:B^(l)→U(n)

{right arrow over (b)}→C({right arrow over (b)})  (1)

[0067] The modulation rate is R=1, if the l input bits are arranged in n symbols, with each symbol comprising l/n bits (e.g. l/n=1 for BPSK and l/n=2 for QPSK).

[0068] The mapping of bit vectors {right arrow over (b)}_(k) onto the symbols transmitted in n stages via an antenna (see FIG. 1) is now replaced by the mapping of bit vectors {right arrow over (b)}_(k) onto a set of space-time symbols ST, all of which are transmitted via n antenna in n stages. The application of these space-time codes is then without influence (transparent) for external transmission blocks, e.g. such as a channel coder.

[0069] At the transmitter the matrix element c_(ijk) of the 2^(l) unitary n×n matrices C_(k) are then processed as space-time variables as follows: if the unitary n×n matrix C_(k) as an ST symbol corresponds to a bit vector {right arrow over (b)}_(k) to be transmitted, a signal corresponding to the matrix element c_(ijk) (with line index i=1, . . . , n, column index j=1, . . . , n) is transmitted from the ith antenna in time window j. With this set-up, the transmission time period for a complete ST symbol is then n time units.

[0070] As all matrices C_(k) are unitary, i.e. C_(k) ^(↑)C_(k)=C_(k)C_(k) ^(↑)=1, the lines and columns of C_(k) are orthonormal. This implies that the power transmitted from each antenna (averaged over time) is identical. Also the total symbol energy (added together over all antennas) is constant in each time slot. It should be noted that the transmission power E_(b) is automatically standardized and independent of n, due to the fact that the C_(k) are unitary.

Channel Model

[0071] A fading channel with simple Rayleigh or Rician distribution is assumed below. Transmission paths (from each antenna) are subject for example to independent Rayleigh fading in each channel state α_(ij) and it may be assumed that the channel does not change significantly during the transmission time of an ST symbol (corresponding to n time slots). Let the signal to noise ratio in each channel be the same, i.e. γ_(b)=E_(b)|N₀.

Receiver

[0072] With the modulation and channel model described above, the receiver will receive a signal at a receiving time, which is offset due to the runtime and corresponds to the transmission time j, said signal resulting from the summation of all matrix elements c_(ijk), j=const., in a matrix column (fixed transmission time j), for a specific code word C_(k), in each instance weighted by a fading factor specific to the channel state, with noise still having to be taken into account. This means, when transmitting a code symbol C_(k), which corresponds to a bit vector {right arrow over (b)}_(k), that the signal (the receiving vector) r_(j) ^(k)=Σ_(i=l) ^(n)c_(ijk)α_(i)+noise is present on the receiver side at a receiving time, which is offset due to the runtime. (In other words: the channel state characterized by α does not depend on the space-time symbol C_(k) sent and is constant for all n time slots.

[0073] Let the index k be omitted below for reasons of clarity.

[0074] If we write the receiving vector as {right arrow over (r)}=(r₁, . . . , r_(n))^(T) and the channel state vector as {right arrow over (a)}=(a₁, . . . , a_(n))∈C^(n), this can be written in matrix form as:

{right arrow over (r)}({right arrow over (b)})=C({right arrow over (b)}){right arrow over (a)}+{right arrow over (z)},  (2)

[0075] where {right arrow over (z)} describes the noise vector distributed as standard. From the point of view of the receiver the receiving symbols {right arrow over (r)}({right arrow over (b)}) are not fixed but are for their part also stochastically distributed variables (i.e. they are dependent on the channel state!). The overall modulation is therefore a mapping of the bits onto the n-dimensional complex number level:

B^(l)→U(n)→C^(n)  (3)

{right arrow over (b)}→C({right arrow over (b)})→{right arrow over (r)}({right arrow over (b)}).  (4)

[0076] For a good code this mapping must of course be uniquely reversible, for each channel state {right arrow over (a)}≠0. This imposes basic conditions on the set of ST symbols {C({right arrow over (b)})}_({right arrow over (b)}∈B) _(^(l)) . In fact for any two symbols C({right arrow over (b)}_(i)) and C({right arrow over (b)}_(j)),{right arrow over (b)}_(i)≠{right arrow over (b)}_(j) the Euclidean interval cannot be zero, i.e. the following must apply:

∥C({right arrow over (b)} _(i)){right arrow over (a)}−C({right arrow over (b)} _(j)){right arrow over (a)}∥ ₂≠0  (5)

[0077] for any state vector {right arrow over (a)}, which is not equal to the zero vector.

[0078] The following must apply in particular:

└C({right arrow over (b)} _(i))−C({right arrow over (b)}_(j))┘•{right arrow over (a)}≠0

[0079] for all {right arrow over (a)}≠0 (zero vector).

[0080] If this condition is satisfied, a unique solution must be defined for the present linear equation system, i.e. det└C({right arrow over (b)}_(i))−C({right arrow over (b)}_(j))┘≠0, where the determinate is equal to the product of the eigenvalues of └C({right arrow over (b)}_(i))−C({right arrow over (b)}_(j))┘.

[0081] Minimizing the equation (4) for all {right arrow over (a)} on condition that ∥{right arrow over (a)}∥=1 gives an eigenvalue equation for the matrix C({right arrow over (b)}_(i))−C({right arrow over (b)}_(j)).

[0082] Thus the minimum interval (determined by the “worst case” channel α″ with the greatest signal distortion) for a defined pair of ST codes is equal to the minimum eigenvalue λ_(min) of C({right arrow over (b)}_(i))−C({right arrow over (b)}_(j)). λ_(min) must be positive so that there is reversibility.

[0083] The following statements are equivalent:

[0084] The eigenvalues λ_(min) of C({right arrow over (b)}_(i))−C({right arrow over (b)}_(j)) are not equal to zero for any pair of ST symbols.

[0085] For all i,j(i≠j) the rank of C({right arrow over (b)}_(i))−C({right arrow over (b)}_(j)) is equal to n. For all i,j(i≠j) the determinant det(C({right arrow over (b)}_(i))−C({right arrow over (b)}_(j))) cannot disappear.

[0086] Given equation (2) an optimum MLD (minimum likelihood detector) minimizes the interval in respect of all possible channel symbols {right arrow over (r)}_(j)=C({right arrow over (b)}_(j))â, where an estimated value â is used for the channel state information (e.g. by pilot sequences): $\hat{b} = {\arg \quad {\min\limits_{j}{{{\overset{\rightarrow}{r} - {{C\left( {\overset{\rightarrow}{b}}_{j} \right)}\hat{\alpha}}}}.}}}$

[0087] A logarithmic likelihood value (“soft symbol”) can be derived for the bit vector {circumflex over (b)} from the interval ∥{right arrow over (r)}−C({right arrow over (b)}_(j))â∥. If mapping of the bits in symbols (Gray Code or similar) is specified (e.g. by pilot sequences) an LLR (log likelihood ratio) value $\log \left( {{L(x)} = \frac{P\left( {b = 0} \right)}{P\left( {b = 1} \right)}} \right)$

[0088] could also be determined for each bit.

[0089] Existence of time-space symbols (ST symbols) with maximum rank

[0090] The question is whether there is a set of 2^(l) ST symbols (matrices C_(k)) for every number n of antennas, with the characteristic that they have a maximum rank. In other words $\begin{matrix} {d_{\min} = {\min\limits_{i \neq j}{{\det \left( {C_{i} - C_{j}} \right)}}}} & (9) \end{matrix}$

[0091] has to be determined for all pairs of C_(i), C_(j).

[0092] In the case of unitary n×n matrices, their rank is always equal to n. The decisive question for the ST codes considered according to the invention now is whether all the differences between the selected matrices have full rank (n).

[0093] If we can construct a set of matrices C={C_(i)} in such a way that they form a group in respect of standard matrix multiplication, things become simpler:

[0094] Lemma 1:

[0095] Let C={C_(i)} be a group. If C_(i)=1 is the unit matrix of the group, then $d_{\min} = {\min\limits_{i \neq 1}{{\det \left( {1 - C_{i}} \right)}}}$

[0096] applies.

[0097] Proof:

[0098] Due to the group characteristic, C_(i) ⁻¹=C_(i) ^(↑∈C. This means that C) _(j)C_(i) ⁻¹=C_(k)∈C and the following applies:

min_(ij)=|det(C _(i−) C _(j))|=min_(ij)|det(1−C _(j) C _(i) ^(↑))|=min_(k)|det(1−C _(k))|.

[0099] Use is also made here of the fact that |det(C)|=1.

EXAMPLE 1

[0100] The Alamouti scheme is based on $c = {\frac{1}{2}\begin{pmatrix} {1 \pm j} & {1 \pm j} \\ {{- 1} \pm j} & {1 \mp j} \end{pmatrix}}$

[0101] etc.

[0102] Actually the Alamouti scheme comprises a subset of the cube group with the order o=24. Eight of the group elements are not used, although they give the same d_(min)=1. In this scheme in principle therefore ld(24)>4 bits could be transmitted without data loss. (However the symbols not transmitted have a poor crest factor, e.g. C=1).

[0103] It is known from the publication “Space Time Codes for High Data Wireless Communication: Performance Criterion and Code Construction” by V. Tarokh referred to above that the rank n of n×n code matrices constructed as space-time block symbols is equal to the degree of diversity. It therefore also follows that the rank of the difference between two n×n code matrices is maximum n. This knowledge is used below.

Complex Symbols

[0104] Theorem 2:

[0105] For any l and n≧2 there is a complex ST modulation scheme with a degree of diversity of n.

[0106] Proof: by Specific Construction

[0107] Note that in this case the ST code {C_(k)({right arrow over (b)}_(k))}_({right arrow over (b)}∈B) _(^(l)) comprises a subset of all possible unitary n×n matrices. On the basis of the spectral theorem set out in Simon, Barry: Representations of Finite and Compact Groups, 1996, Graduate Studies in Mathematics, American Mathematical Society, ISBN 0-8218-0453-7, every unitary matrix C can be written as: $C = {{V\begin{pmatrix} \lambda_{1} & \quad & 0 \\ \quad & \cdots & \quad \\ 0 & \quad & \lambda_{n} \end{pmatrix}}V^{- 1}}$

[0108] where V is unitary and λ_(i)=exp(jβ_(i)) (in the argument for the function exp, j is the imaginary unit). Let us now select ${\beta_{i} = {\frac{2\pi}{2^{l}}q_{i}}},$

[0109] where q_(i) is any odd whole number. ThenC² ^(l) =1 applies and {C_(k)}_(k=. . . 2) _(^(l)) ⁻¹ is an (Abelian) group, with a generator $\begin{matrix} {C = {{V\begin{pmatrix} {\exp \left( {\frac{2\pi \quad j}{2^{l}}q_{1}} \right)} & \quad & 0 \\ \quad & \cdots & \quad \\ 0 & \quad & {\exp \frac{2\pi \quad j}{2^{l}}q_{n}} \end{pmatrix}}V^{- 1}}} & (7) \end{matrix}$

[0110] for any fixed V. The eigenvalues of C_(k) are $\lambda_{i}^{k} = {{\exp \left( {\frac{2\pi \quad j}{2^{l}}q_{i}^{k}} \right)}.}$

[0111] Constructing with k=0 . . . 2¹−1 means that they are clearly different from 1. Based on this it can be shown that d_(min)=min_(k≠0) |det(1−C_(k))≠0.

[0112] As |det(V)|=1, $d_{\min} = {{\min_{k \neq 0}{{\det \left( {1 - \begin{pmatrix} \lambda_{1} & \quad & 0 \\ \quad & \cdots & \quad \\ 0 & \quad & \lambda_{n} \end{pmatrix}^{k}} \right)}}} = {{\min_{k \neq 0}{{\prod\limits_{i - 1}^{n}\quad \left( {1 - \lambda_{i}^{k}} \right)}}} \neq 0}}$

[0113] Note that the key to the above proof is based on the fact that it is always possible to construct a set of unitary matrices, the eigenvalues of which are all different from 1.

[0114] Equation (7) therefore provides the basis for a method for constructing complex unitary n×n code matrices C_(k), which can be used as symbol words for space-time block codes for any number of n≧2 transmitter antennas, with said space-time block codes providing a maximum diversity n, as the rank of the complex unitary n×n code matrices C_(k) is equal to n.

[0115] If a group of code matrices is calculated according to the above equation (7), the result is a list of 2^(l) unitary n×n matrices with complex matrix elements, which can be used as a basis for a non-linear space-time block code, with which a radio transmission link shown diagrammatically in FIG. 2 can be operated in a digital mobile radio network. This knowledge is based on the technical doctrine of Claim 1.

[0116] In the case of n=3 transmitter antennas and BPSK modulation, i.e. number of transmitter antennas n=length of the symbol words l, for the sake of clarity a specific example of such a list of code matrices C_(k) is given in the reference table in FIG. 3.

[0117] In the reference table shown in FIG. 1 the bits b_(i ∈{0,1} are mapped onto) 2³=8 bit vectors. These are specifically the eight possible vectors (0,0,0); (0,0,1); (0,1,1); (0,1,0); (1,1,0); (1,1,1); (1,0,1); (1,0,0) formed by permutation.

[0118] These eight bit vectors are each uniquely (reversibly uniquely) associated with one of eight code matrices C_(k)=C(:;:k). Here the code matrices in table 1 are unitary complex 3×3 code matrices, the matrix elements of which are calculated according to equation (7).

[0119] If the association between the eight bit vectors and the eight code matrices C_(k), shown as an example in FIG. 3, is used for example with a digital mobile radio system as shown in FIG. 2, and a bit vector (1,0,1) for example arrives there at the ST coder, said coder transmits each column of the code matrix C₇ in three successive time slots, so that the complex matrix element c₁₁₇=0,0000+0.6442i is transmitted via the antenna 1 in the time slot 1, the complex matrix element c₂₁₇=−0.2175−0.0412i via the antenna 2 in the time slot 1, etc. until the complex matrix element C₃₃₇=0.0000−0.5065i is transmitted via the antenna 3 in the time slot 3 (in FIG. 31 is used for the imaginary unit).

[0120] On the receiver side the complete code matrix C₇ is then reconstructed from the signals received by a receiver antenna Rx as described above by means of an MLD detector and the original bit vector (1,0,1) is associated with this again in a reverse mapping.

[0121] For practical purposes only the matrices calculated according to equation (7) should be stored with the respective bit vector association in a reference table in the form of the table shown in FIG. 3 in a storage unit of the ST coder on the transmitter side and correspondingly an identical reference table in a receiver-side ST decoder.

[0122] The transmitter-side ST coder can be integrated in a base station of a digital mobile radio network and the receiver-side ST decoder in the mobile station of a digital mobile radio network. In principle this can however be reversed.

[0123] The reference tables can be stored as computer program products in machine-readable form, for example on diskette or in the form of machine-readable files, which can be transmitted via the internet or the radio transmission links and if necessary can be input into corresponding storage units of transmitter-side ST coders or receiver-side ST decoders in the base stations or mobile stations in a digital mobile radio network.

Real Symbols

[0124] For real symbols the characteristic used above for constructive proof resulting in equation (7), that it is always possible to construct a set of unitary matrices, the eigenvalues of which are all different from 1, no longer applies.

Real ST Codes

[0125] If we restrict the ST code matrices to real matrix elements c_(ijk), a maximum rank (and therefore a maximum order of diversity of the ST codes) can only be constructed for the case of an even number of antennas.

[0126] Theorem 3:

[0127] Real ST codes of the order 2n+1 have a non-maximum diversity order.

[0128] Proof:

[0129] Any O∈SO(2n+1) can be written as O=VDV⁻¹(8) (see Simon, Barry: Representations of Finite and Compact Groups, 1996, Graduate Studies in Mathematics, American Mathematical Society, ISBN 0-8218-0453-7),

[0130] where V is orthogonal and $D = \begin{pmatrix} \begin{pmatrix} {\cos \quad \Phi_{1}} & {{- \sin}\quad \Phi_{1}} \\ {\sin \quad \Phi_{1}} & {\cos \quad \Phi_{1}} \end{pmatrix} & \quad & \quad & 0 \\ \quad & \begin{pmatrix} {\cos \quad \Phi_{2}} & {{- \sin}\quad \Phi_{2}} \\ {\sin \quad \Phi_{2}} & {\cos \quad \Phi_{2}} \end{pmatrix} & \quad & \quad \\ \quad & \quad & {\quad \cdots} & \quad \\ {0\quad} & \quad & \quad & 1 \end{pmatrix}$

[0131] (For matrices with detO=−1 the proof is essentially identical).

[0132] Let us now consider det(O₁−O₂)=det(1−O₂O₁ ⁻¹)=det(1−O_(2l)), where due to the group structure O_(2l)∈SO(2n+1) applies. O_(2l) then has the structure according to equation (8) and the determinate det(O₁−O₂) disappears.

[0133] The reason why SO(2n+1) does not provide a maximum diversity order is based on the fact that every orthogonal matrix with an uneven dimension has (at least) one eigenvalue equal to 1.

[0134] For an even number of antennas, the additional 1 does not occur at the position (n,n) of D and a code construction similar to the unitary one is possible.

ST Symbol Optimization

[0135] Although the theorem for complex ST modulation is constructive, it does not provide an optimal ST code. The asymptomatic symbol error is of the form ${P_{e} \sim {c\quad \left( \frac{E_{b}}{N_{0}} \right)^{- n}}},$

[0136] as a result of which an optimum diversity order occurs for $\frac{E_{b}}{N_{0}}.$

[0137] However the constant c is not minimal.

[0138] Practical methods for code construction based on optimization considerations are therefore set out below. The results are confirmed by simulation.

[0139] In the sections below, construction methods are set out for “good” unitary ST codes, i.e. those ST codes with which the intervals between the code symbols are optimized.

Optimization

[0140] The idea is to find suitable parameterization for a set of unitary U(n) matrices and then to minimize a suitable metric numerically, said metric representing the intervals between the code words. As the distance measurement D_(ev)=min {eigenvalues of C({right arrow over (β)}_(i))−C{right arrow over (β)}_(j))} cannot be differentiated, we select

d _(ij) :=d(C({right arrow over (β)}_(i)),C({right arrow over (β)}_(j)))=|det(C({right arrow over (β)}_(i))−C({right arrow over (β)}_(j)))|.

[0141] Here {right arrow over (β)}_(k) stands for the parameter of the kth code matrix C_(k). As a target functional $\begin{matrix} {{E_{q}\left\lbrack {{\overset{\rightarrow}{\beta}}_{1},{\overset{\rightarrow}{\beta}}_{2},\ldots}\quad \right\rbrack} = \left( {\sum\limits_{i < j}^{2^{l}}\quad \left\lbrack {d\left( {{C\left( {\overset{\rightarrow}{\beta}}_{i} \right)},{C\left( {\overset{\rightarrow}{\beta}}_{j} \right)}} \right)} \right\rbrack} \right)^{\frac{1}{q}}} & (6) \end{matrix}$

[0142] (6)

[0143] for global extremal value creation, we can use the L_(q) standard of all both-way code intervals. For example, q→−∞ gives the minimum standard (a large negative q can be used for numerical optimization); the case where q=−1 can be interpreted as electrical potential. In fact due to the compactness of U(n) the problem is comparable to the minimization of the electrical energy from 2^(l) equally charged particles moving on a sphere. Positive values for q are not meaningful, as they exclude no distances which could be zero (i.e. two particles, at the same place, do not generate infinite energy, so they would not repel each other).

The Case of the Group SU(2)

[0144] This case corresponds to n=2 antennas.

[0145] In this case the energy function described above can be specifically constructed, as according to Simon Barry, “Representation of Finite and Compact Groups”, Graduate Studies in Mathematics, Volume 10, American Mathematical Society, every unitary matrix C can be parameterized as

C=1β₀ +j(β₁σ₁+β₂σ_(z)+β₃σ₃):={right arrow over (β)}•{right arrow over (σ)}

[0146] where {right arrow over (σ)}_(i) are the known Pauli spin matrices: ${{\overset{\rightarrow}{\sigma}}_{1}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}},{{\overset{\rightarrow}{\sigma}}_{2}\begin{pmatrix} 0 & {- j} \\ j & 0 \end{pmatrix}},{{\overset{\rightarrow}{\sigma}}_{3}\begin{pmatrix} 1 & 0 \\ 0 & {- 1} \end{pmatrix}},$

[0147] and here j is the imaginary unit.

[0148] The real parameters β here are subject to the restriction ${\sum\limits_{i = 0}^{3}\quad \beta_{i}^{2}} = 1.$

[0149] In fact this ensures that SU(2) is isomorphous in relation to a 3-sphere (a sphere with four dimensions). It can easily be shown that det(C({right arrow over (β)}hd i)−C({right arrow over (β)}_(j)))=2−2{right arrow over (β)}_(i)•{right arrow over (β)}_(j), therefore we define

d _(ij)={square root}{square root over (1−β)}_(i)•{right arrow over (β)}_(j)

[0150] as the measure of distance between two code matrices. Note that d_(ij) is actually a metric in the case of SU(2).

[0151] We use d=Σ_(i<j)d_(ij) as the full interval. A 3-sphere is parameterized simply by three angles: ${\overset{\rightarrow}{\beta}}_{i} = \begin{pmatrix} {\sin \quad \Phi_{i3}} & {\sin \quad \Phi_{i2}} & {\sin \quad \Phi_{il}} \\ {\cos \quad \Phi_{i3}} & {\sin \quad \Phi_{i2}} & {\sin \quad \Phi_{il}} \\ \quad & {\cos \quad \Phi_{i2}} & {\sin \quad \Phi_{il}} \\ \quad & \quad & {\cos \quad \Phi_{il}} \end{pmatrix}$

[0152] The gradient here is: ${\frac{\partial\quad}{\partial\Phi_{kl}}d} = {{- \left( {\frac{\partial\quad}{\partial\Phi_{kl}}{\overset{\rightarrow}{\beta}}_{k}} \right)} \cdot {\sum\limits_{i \neq k}^{\quad}\quad \frac{{\overset{\rightarrow}{\beta}}_{i}}{d_{ik}}}}$

[0153] Optimization methods based on that of the steepest descent provide fast results for a reasonable 1.

[0154] The resulting code spectrum for QPSK (l=4) is compared in FIG. 5 with the code spectrum for the Alamouti scheme. As can be seen, the minimum interval with this procedure is greater than with the Alamouti scheme. Therefore in the asymptomatic extreme case it should be anticipated that this non-linear code will show a higher coding gain (same bit error rate for lower signal to noise ratio).

SU(3) and an Implicit Numerical Gradient Method

[0155] In cases with more than two transmitter antennas, i.e. SU(n), n>2, the specific calculation of determinants becomes very extensive. For gradient methods (e.g. the conjugate gradient method, see W. Press, B. Flannery, S. Tenolsky, W. Vetterling: “Numerical recipes in C”, Cambridge University Press, ISBN 0-521-35465-X) is it however sufficient to calculate the local gradient.

[0156] We define the L_(m) interval of the ith matrix in respect of all others as: $\begin{matrix} {d_{i}^{m} = {{\sum\limits_{j}^{\quad}\quad {{\det \left( {C_{j} - C_{i}} \right)}}^{m}} = {{\sum\limits_{j}^{\quad}\quad {{\det \left( {1 - {C_{j}^{\dagger}C_{i}}} \right)}}^{m}}:={\sum\limits_{j}^{\quad}\quad {{\det \left( {1 - A_{ij}} \right)}}^{m}}}}} & (9) \end{matrix}$

[0157] (9)

[0158] If the ith code matrix is varied by an infinitesimal (unitary) rotation

C_(i)→C_(i)exp(j{right arrow over (σ)}•{right arrow over (δ)}_(i))≈C_(i)+j{right arrow over (δ)}_(i)•C_(i){right arrow over (σ)}

[0159] the following results for the gradient $\begin{matrix} {{{\overset{\rightarrow}{\nabla}}_{\delta_{i}}d_{i}^{m}}:={{\overset{\rightarrow}{g}}_{i} = {n{\sum\limits_{j \neq i}^{\quad}\quad {{d_{ij}}^{n}\quad {Re}\quad {{Tr}\left( {A_{ij}^{- 1}{\overset{\rightarrow}{B}}_{ij}} \right)}}}}}} & (10) \end{matrix}$

[0160] where:

{right arrow over (B)} _(ij) =−jC _(j) ^(†) C _(i){right arrow over (σ)}.

[0161] The σ_(i) here are the corresponding Hermitian standard spin matrices. For larger-dimensional spaces (n>2) these are specified for example in the book “Gauge theory of elementary particle physics” by Ta-Pei Cheng and Ling-Fong Li.

[0162] A variation with an increment δ is then applied to the ith code word according to

C_(i)→C_(i)exp(jδ{right arrow over (σ)}•{right arrow over (g)}_(i))  (11)

[0163] An algorithm with the steepest descent then functions as follows:

[0164] 1) Generate a random quantity of 2^(l)unitary n×n matrices S_(k),k=1 . . . 2^(l) as initial matrices (this can also be done according to equation (7)).

[0165] 2) Calculate the gradient vectors according to equation (10).

[0166] 3) “Rotate” the matrices according to the equation (11), then iterate according to stage 2).

[0167] Naturally a conjugate gradient method can be constructed correspondingly and stochastic gradient methods are also possible to find the global extreme value creator.

[0168] An example of 3 antennas and a QPSK modulation, which provides 2⁶=64 ST matrices, is shown in FIG. 6. FIG. 6 shows a spectrum for SU(3), i.e. three transmitter antennas, and QPSK modulation using a minimum standard. If the same method is used for SU(3) but for a BPSK modulation and using different standards (min, L⁻¹, L⁻²), the spectra shown in FIG. 7 are obtained.

[0169] It seems to be worthy of note that it is possible to find eight code matrices in SU(3), which all show the same mutual interval. This corresponds to a tetrahedron in the standard three-dimensional space.

[0170]FIG. 4 shows a two-dimensional illustration of this. The eight bit vectors (0,0,0); (0,0,1); (0,1,1); (0,1,0); (1,1,0); (1,1,1); (1,0,1); (1,0,0) are mapped onto eight code matrices (e.g. the code matrices specifically set out in FIG. 3), which have optimum intervals in respect of each other.

[0171]FIG. 8 shows a spectrum for n=3 antennas and BPSK using different optimization criteria.

[0172] In addition to the numerical optimization method described above, there are further approaches for code optimization, i.e. the specific construction of ST symbols in the form of unitary n×n matrices with optimized intervals in respect of each other:

[0173] Construction of Hyperspheres:

[0174] Let a (hyper) sphere with a defined radius be constructed around a defined code symbol (e.g. a unitary n×n matrix C_(k)), a second code symbol be found on the sphere around this first code symbol and a third symbol be constructed as the point of intersection of the (hyper) spheres and the first and second code symbol. Then let further code symbols be constructed correspondingly and iteratively as the points of intersection of further (hyper) spheres around the code words already found in each instance.

[0175] In the event that the code words are unitary n×n matrices with n≧2, a “sphere” with radius r around a code word is defined by

S _(r) ={C′|det(C′−C)=r}.

[0176] It can be constructed by

r=det(C−C′)=det(1−C ^(†) C′)=det(1−exp(j{right arrow over (σ)}{right arrow over (β)}))=Π_(i=1) ^(n)(1−exp(λ_(i))),

[0177] where λ_(i), are the eigenvalues of j{right arrow over (σ)}{right arrow over (β)},

[0178] For example it is found in SU(2) that

r=4 sin²(1/2 {square root}{square root over (β₁ ²+β₂ ²+₃ ² )}).

[0179] This means that such a sphere can be parameterized as C′=C exp(j{right arrow over (σ)}{right arrow over (β)}) with restriction of the sum of the squared βs.

[0180] A sphere with radius 1 is for example defined by

S ₁(C)={C exp(j{right arrow over (σ)}{right arrow over (β)})|β₁ ²+β₁ ²+β₁ ²=π²/9,−π/3≦β_(i)≦π/3}.

[0181] Using this idea, it is actually possible to construct the Alamouti scheme (by constructing two spheres with the radius 1 around the elements 1 and −1 (the center of SU(2), . . . Z(SU(2)).

[0182] This gives the following formula for the code words: ${C = {\exp \left( {j\frac{\pi}{3\sqrt{3}}\left( {1 + b_{0}} \right){\sum\limits_{i = 1}^{3}\quad {\left( {1 - {2b_{i}}} \right)\sigma_{i}}}} \right)}},$

[0183] where b_(i)∈{0,1) are the bits and σ_(I) the Hermitian standard spin matrices.

[0184] Specific calculation leads back to the already known Alamouti scheme, thereby proving the correctness of the approach of construction by means of hyperspheres.

[0185] As the example of the Alamouti scheme in conjunction with FIG. 5 shows, a local optimum can be found using the hypersphere method but not necessarily a global optimum.

[0186] Nevertheless, for n>2 the eigenvalues of j{right arrow over (σ)}{right arrow over (β)} are (analytically) very extensive.

[0187] Therefore a different parameterization seems more successful here for n>2:

[0188]r=det(1−C ^(†) C′)=det(1−VDV^(†))=det(1−D)=Π_(i)(1-exp(jλ_(i)))

[0189] If the eigenvalues λ_(i) are kept constant, the result is a sphere around the code symbol C with

C′=CVDV^(↑.)

[0190] V can be parameterized here as V=exp(j{right arrow over (ρ)}{right arrow over (β)}).

Unitary Representations of Finite Groups

[0191] A further method of code construction is based on the use of finite groups. For this, see also Simon Barry, “Representation of Finite and Compact Groups”, Graduate Studies in Mathematics, Volume 10, American Mathematical Society.

[0192] The combination of two elements in a finite group leads back to one element of the group, as the group is closed after a group axiom. Also the multiplication of two unitary matrices again provides a unitary matrix. There are correspondingly numerous representations of finite groups, in which a unitary matrix is associated with each group element. If such a representation of a finite group, in which the number of group elements is greater than 2^(l), is selected, good initial values are obtained for the optimization methods referred to above.

[0193] For this, it is necessary to try to find the unitary representations of finite groups (with the dimension n), in which o(G)≧2^(l). There is no guarantee that this produces optimum results.

[0194] Some Simulation Results:

[0195] Simulations are carried out for the instances with n=2 and n=3 antennas, in which only BPSK has (currently) been used. Theoretical limits for antenna diversity can be derived in closed form (see also for example J. Proakis, M. Salehi: “Communications Systems Engineering”, Prentice Hall Int., ISBN 0-13-300625-5, 1994). These can be seen as continuous lines in FIG. 9. Simulation results are shown together with 70% confidence intervals.

[0196] As shown in FIG. 10, the theoretical limit for high E_(b)/N₀ in the case of three antennas is reached for the L_(min) code. This is due to the fact that L_(min) is actually the maximum possible smallest interval. For a good signal to noise ratio in practice only the errors at the smallest intervals make a contribution.

[0197] However with a low signal to noise ratio the characteristics of the code deteriorate and they can even be worse than with a diversity of two antennas. In fact ST codes can be seen as a higher modulation scheme (resulting in expansion of the symbol space). It is of course not possible to increase the number of symbols without reducing the intervals in a compact space. This problem becomes even more significant for QPSK (with 64 ST symbols).

[0198] For the L⁻¹ code performance is better in the areas with a low signal to noise ratio. Gray coding was used for this code (which has two different intervals in the minimum eigenvalue standard), in order to avoid multiple bit errors in the event of a symbol error. As can be seen in the spectrum below, each code matrix has precisely two immediate neighbors. All other code words have a larger interval. As can be seen, the code is close to the theoretical limit for E_(b)/N₀>4 dB and it outperforms the L_(min) code in the area with low E_(b)/N₀. 

1. Method for operating a digital mobile radio network with orthogonally structured space-time block transmission codes with maximum diversity n×m for a defined number of n transmitter antennas and m,m=2, 3,
 4. . . , receiver antennas, with a transmitting station of the mobile radio network being provided with n,n=2, 3,
 4. . . , transmitter antennas and a set of 2′ data bit vectors {right arrow over (b)}=(b₁, b₂. . . b_(l))∈ B^(l) with 1 bits b_(i∈{)0,1} to be transmitted from this transmitting station being mapped by a one-to-one mapping STM:B^(l)− > U(n) $\quad {\overset{\rightharpoonup}{b}->{C\left( \overset{\rightharpoonup}{b} \right)}}$

onto a set of 2′ space-time symbols C_(k), k=0, 2, . . . , 2¹−1, with each code symbol C_(k) corresponding to a unitary n×n matrix and the matrix elements c_(ijk) of each of the 2′ code symbols C_(k) being interpreted as space-time variables so that on transmission of one of the 2′ code symbols C_(k) for each of its matrix elements c_(ijk) a corresponding signal is transmitted from the transmitter antenna i in a time interval j via a fading channel associated with the transmitter antenna i, so that signals transmitted in the time interval j and corresponding to the matrix elements c_(ijk), i=1, . . . , n of the code symbol C_(k) are received at each of m receiver antennas of a receiving station within the range of the transmitting station and with a corresponding data bit vector {right arrow over (b)}∈B^(l) to be transmitted being decoded by the reverse mapping SMT⁻¹:U(n)− > B^(l) $\quad {{C\left( \overset{\rightharpoonup}{b} \right)}->\overset{\rightharpoonup}{b}}$

due to the orthogonal structure of the space-time block transmission code and by decoupling the signals transmitted by the n transmitter antennas and corresponding to the matrix elements c_(ijk), characterized in that the matrix elements c_(ijk) corresponding to the signals to be transmitted are the elements of 2¹ unitary n×n matrices C_(k), k=0, 2, . . . , 2¹−1, constructed acc ording to the following specification: $C_{k} = {\begin{pmatrix} c_{11k} & \quad & c_{1{nk}} \\ \quad & \cdots & \quad \\ c_{n1k} & \quad & c_{nnk} \end{pmatrix} = {{V\begin{pmatrix} {\exp \left( {\frac{2\pi \quad j}{2^{l}}q_{1}k} \right)} & \quad & 0 \\ \quad & \cdots & \quad \\ 0 & \quad & {\exp \frac{2\pi \quad j}{2^{l}}q_{1}k} \end{pmatrix}}V^{- 1}}}$

(j is the imaginary unit here) where V is any unitary complex n×n matrix, C_(1jk) the generally complex matrix elements of C_(k), j the imaginary unit, q_(i), I=1, . . . , n any odd whole numbers, and k=0, 1, . . . , 2¹−1.
 2. Method according to claim 1, characterized in that n is even and c_(ijk) are real.
 3. Method for operating a digital mobile redio network with orthogonally structured space-time block transmission codes with maximum diversity n×m for a defined number of n transmitter antennas and m,m=2, 3, 4 . . . , receiver antennas, with a transmitting station in the mobile radio network being provided with n,n=2, 3, 4 . . . , transmitter antennas and a set of 2¹ data bit vectors {right arrow over (b)}=(b₁, b₂. . . b_(l))∈B^(l) with 1 bits b₁ ∈{0,1} to be transmitted from this transmitting station being mapped by a one-to-one mapping STM:B^(l)− > U(n) $\quad {\overset{\rightharpoonup}{b}->{C\left( \overset{\rightharpoonup}{b} \right)}}$

onto a set of 2′ space-time symbols C_(k), k=0, 2, . . . , 2¹−1, with each code symbol C_(k) corresponding to a unitary n×n matrix and the matrix elements c_(ijk) of each of the 2′ code symbols C_(k) being interpreted as space-time variables so that on transmission of one of the 2′ code symbols C_(k) for each of its matrix elements c_(ijk) a corresponding signal is transmitted from the transmitter antenna i in a time interval j via a fading channel associated with the transmitter antenna i, so that signals transmitted in the time interval j and corresponding to the matrix elements c_(ijk), i=1, . . . , n of the code symbol C_(k) are received at each of m receiver antennas of a receiving station within the range of the trasmitter station and with a corresponding data bit vector {right arrow over (b)}∈B^(l) to be transmitted being decoded by the reverse mapping SMT⁻¹:U(n)− > B^(l) $\quad {{C\left( \overset{\rightharpoonup}{b} \right)}->\overset{\rightharpoonup}{b}}$

due to the orthogonal structure of the space-time block transmission code and by decoupling the signals transmitted by the n transmitter antennas and corresponding to the matrix elements c_(ijk), characterized in that the matrix elements c_(ijk) corresponding to the signal to be transmitted are the elements of 2¹ unitary n×n matrices C_(k), k=0, 2, . . . , 2 ¹−1, optimized numerically according to the following specification: a) an initial set of 2¹ unitary n×n initial matrices S_(k), k=0, 2, . . . , 2¹−1 are generated numerically at random; b) these initial matrices S_(k), k=0, 2, . . . , 2¹−1 are parameterized; c) The variable d_(ij):=d (S({right arrow over (β)}_(i)), S({right arrow over (β)}_(j)))=|det(S({right arrow over (β)}_(i))−S({right arrow over (β)}_(j)))| is selected as the measure of distance between two initial matrices, with {right arrow over (β)}_(i) representing the parameters of the ith initial matrix; d) the target functional ${E_{q}\left\lbrack {{\overset{\rightarrow}{\beta}}_{1},{\overset{\rightarrow}{\beta}}_{2},\ldots}\quad \right\rbrack} = \left( {\sum\limits_{i < j}^{2^{l}}\quad \left\lbrack {d\left( {{S\left( {\overset{\rightarrow}{\beta}}_{i} \right)},{S\left( {\overset{\rightarrow}{\beta}}_{j} \right)}} \right)} \right\rbrack} \right)^{\frac{1}{q}}$

 is minimized by numerical variation of the parameter {right arrow over (β)}_(i); d) the n×n matrices, for the parameter {right arrow over (β)}_(i) of which the target functional ${E_{q}\left\lbrack {{\overset{\rightarrow}{\beta}}_{1},{\overset{\rightarrow}{\beta}}_{2},\ldots}\quad \right\rbrack} = \left( {\sum\limits_{i < j}^{2^{l}}\quad \left\lbrack {d\left( {{C\left( {\overset{\rightarrow}{\beta}}_{i} \right)},{C\left( {\overset{\rightarrow}{\beta}}_{j} \right)}} \right)} \right\rbrack} \right)^{\frac{1}{q}}$

 was minimized numerically, are selected as final n×n matrices, which correspond in each instance to a code symbol C_(k), and their matrix elements are selected as the matrix elements c_(ijk) corresponding to the signals to be transmitted.
 4. Method according to claim 3, characterized in that n=2 and a) that every matrix S_(k) of the initial set of unitary n×n initial matrices is parameterized as S=1β₀ +i(β₁σ₁+β₂σ₂+β₃σ₃)=β·σ  where the following applies for σ_(I): ${\sigma_{1}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}},{\sigma_{2}\begin{pmatrix} 0 & {- j} \\ j & 0 \end{pmatrix}},{\sigma_{3}\begin{pmatrix} 1 & 0 \\ 0 & {- 1} \end{pmatrix}},$

and j is the imaginary unit, and the real parameters β are subject to the following restriction: ${{\sum\limits_{i = 0}^{3}\quad \beta_{i}^{2}} = 1},$

b) that d_(ij)={square root}{square root over (1−β)}_(i)·{right arrow over (β)}_(j) is selected as the measure of distance between two matrices A_(i), A_(j); c) that the parameter vectors {right arrow over (β)}_(i) are parameterized as ${{\overset{\rightarrow}{\beta}}_{i} = \begin{pmatrix} {\sin \quad \Phi_{i3}} & {\sin \quad \Phi_{i2}} & {\sin \quad \Phi_{il}} \\ {\cos \quad \Phi_{i3}} & {\sin \quad \Phi_{i2}} & {\sin \quad \Phi_{il}} \\ \quad & {\cos \quad \Phi_{i2}} & {\sin \quad \Phi_{il}} \\ \quad & \quad & {\cos \quad \Phi_{il}} \end{pmatrix}},$

d) that the gradients ${{\frac{\partial}{\partial\Phi_{k1}}d} = {{- \left( {\frac{\partial}{\partial\Phi_{kl}}{\overset{\rightarrow}{\beta}}_{k}} \right)}\bullet {\sum\limits_{i \neq k}\frac{{\overset{\rightarrow}{\beta}}_{i}}{d_{ik}}}}}\quad$

 are minimized numerically for all matrices of the initial set of n×n matrices by iteration; e) that the n×n matrices corresponding to the minimized gradients are selected as final n×n matrices, which correspond in each instance to a code symbol C_(k), and their matrix elements are selected as the matrix elements c_(ijk) corresponding to the signals to be transmitted.
 5. Method according to claim 3, characterized in that a) an initial set of 2¹ unitary n×initial matrices S_(k), k=0, 2, . . . , 2¹−1 is generated numerically at random; b) the L_(m) interval of the ith initial matrix in respect of all others is defined as: ${d_{i}^{m} = {{\sum\limits_{j}\quad {{\det \left( {S_{j} - S_{i}} \right)}}^{m}} = {{\sum\limits_{j}\quad {{\det \left( {1 - {S_{j}^{\dagger}S_{i}}} \right)}}^{m}}:={\sum\limits_{j}\quad {{\det \left( {1 - A_{ij}} \right)}}^{m}}}}};$

c) the gradients ${{\overset{\rightarrow}{\nabla}}_{\delta_{i}}d_{i}^{m}}:={{\overset{\rightarrow}{g}}_{i} = {n{\sum\limits_{j \neq i}{{d_{ij}}^{n}{Re}\quad {{Tr}\left( {A_{ij}^{- 1}{\overset{\rightarrow}{B}}_{ij}} \right)}}}}}$

 are calculated, with {right arrow over (B)}_(ij=−jS) _(j) ^(t)S_(i){right arrow over (σ)} and σ_(i) representing the corresponding Hermitian standard spin matrices; d) the ith code matrix S_(i) is varied by an infinitesimal (unitary) rotation S _(i) →S _(i) exp(j{right arrow over (σ)}·{right arrow over (δ)}_(i))≈S _(i) +j{right arrow over (δ)} _(i) ·S _(i){right arrow over (δ)} e) the gradients ${{\overset{\rightarrow}{\nabla}}_{\delta_{i}}d_{i}^{m}}:={{\overset{\rightarrow}{g}}_{i} = {n{\sum\limits_{j \neq i}{{d_{ij}}^{n}{Re}\quad {{Tr}\left( {A_{ij}^{- 1}{\overset{\rightarrow}{B}}_{ij}} \right)}}}}}$

 are calculated by iterative calculation, until they are minimized, f) the n×n matrices corresponding to the minimized gradients are selected as final n×n matrices, which correspond in each instance to a code symbol C_(k), and their matrix elements are selected as the matrix elements c_(ijk) corresponding to the signals to be transmitted.
 6. Method according to one of claims 3 to 5 characterized in that the initial matrices S_(k) are calculated according to the following scheme: $S_{k} = {\begin{pmatrix} s_{11k} & \quad & s_{1{nk}} \\ \quad & \cdots & \quad \\ s_{n1k} & \quad & s_{nnk} \end{pmatrix} = {{V\begin{pmatrix} {\exp \left( {\frac{2\pi \quad j}{2^{l}}q_{1}k} \right)} & \quad & 0 \\ \quad & \cdots & \quad \\ 0 & \quad & {\exp \frac{2\pi \quad j}{2^{l}}q_{1}k} \end{pmatrix}}V^{- 1}}}$

(j is the imaginary unit here) V is any unitary complex n×n matrix, S_(ijk) the generally complex matrix elements of S_(k), q_(i), i=1, . . . , n any odd whole numbers, and k=0, 1, . . . , 2¹−1.
 7. Method according to one of claims 3 to 5, characterized in that a mapping of a finite group G with the dimension n and the order o(G)≧2¹ onto unitary n×n matrices is determined, said matrices then being used as initial matrices S_(k).
 8. Method for operating a digital mobile radio network with orthogonally structrued space-time block transmission codes with maximum diversity n×m for a defined number of n trasnmitter antennas and m,m=b 2, 3, 4 . . . , receiver antennas, with a transmitting station of the mobile radio network being provide with n,n=2, 3, 4 . . . , transmitter antennas and a set of 2′ data bit vectors {right arrow over (b)}=(b₁ , b₂ . . . b_(l))∈B^(l) with 1 bits b_(i) ∈{0,1} to be transmitted from this transmitting station being mapped by a one-to-one mapping STM:  B^(l) → U(n) $\quad \left. \overset{\rightharpoonup}{b}\rightarrow{C\left( \overset{\rightharpoonup}{b} \right)} \right.$

onto a set of 2′ space-time symbols C_(k), k=0, 2, . . . , 2¹−1, with each code symbol C_(k) corresponding to a unitary n×n matrix and the matrix elements c_(ijk) of each of the 2′ code symbols C_(k) being interpreted as space-time variables so that on transmission of one of the 2′ code symbols C_(k) for each of its matrix elements c_(ijl) a corresponding signal is transmitted from the transmitter antenna i in a time interval j via a fading channel associated with the transmitter antenna i, so that signals transmitted in the time interval j and corresponding to the matrix elements c_(ijk), i'21, . . . , n of the code symbol C_(k) are received at each of m receiver antennas of a receiving station within the range of the transmitting station and with a corresponding data bit vector {right arrow over (b)}∈B^(l) to be transmitted being decoded by the reverse mapping STM⁻¹:  U(n) → B^(l) $\quad \left. {C\left( \overset{\rightarrow}{b} \right)}\rightarrow\overset{\rightarrow}{b} \right.$

due to the orthogonal atructure of the space-time block transmission code and by decoupling the signals transmitted by the n transmitter antennas and corresponding to the matrix elements c_(ijk), characterized in that the matrix elements c_(ijk) corresponding to the signals to be transmitted are the elements of 2¹ unitary n×n matrices C_(k), k=0, 2, . . . , 2¹−1, constructed according to the following specification: a) a “hypersphere” with a radius r is constructed numerically around any first code symbol C₁ represented by a unitary n×n matrix, i.e. the quantity of all code symbols is determined, for which the following applies: S_(r)={C′|det(C′−C_(i))=r}, which results from calculating ${r = {{\det \left( {C - C^{\prime}} \right)} = {{\det \left( {1 - {C_{1}^{\dagger}C^{\prime}}} \right)} = {{\det \left( {1 - {\exp \left( {j\quad \overset{\rightarrow}{\sigma}\quad \overset{\rightarrow}{\beta}} \right)}} \right)} = {\prod\limits_{i = 1}^{n}\left( {1 - {\exp \left( \lambda_{i} \right)}} \right)}}}}},$

 where λ_(i the eigenvalues of j{right arrow over (α)}e,rar β;) b) On this hypersphere a second code symbol C₂ represented by a unitary n×n matrix is selected, around which a hypersphere is again constructed in a similar manner to stage a); c) the point of intersection of the two hyperspheres around the code symbols C₁, C₂ has produced at least one further code symbol C₃, around which further hyperspheres are iteratively constructed as set out in stage a), until a full set of 2¹ code symbols C_(k) constructed in this way and represented in each instance by a unitary n×n matrix has been constructed.
 9. Base station, which can be used in a digital mobile radio network in a method according to one of the preceding claims 1 to 8 and which contains a storage unit, in which the associationof individual bit vectors to be transmitted with space-time block symbols is stored in the form of association tables, which contain the matrix elements c_(ijk) used in one of the methods according to claims 1 to
 8. 10. Mobile station, which can be used in a digital mobile radio network in a method according to one of the preceding claims 1 to 8 and which contains a storage unit, in which the association of individual bit vectors to be transmitted with space-time block symbols is stored in the form of association tables, which contain the matrix elements c_(ijk) used in one of the methods according to claims 1 to
 8. 11. Computer program product, stored on a computer-readable medium, with the computer program product containg machine-readable program tools for reading in association tables contained in the computer produce in a storage unit of a base station or a mobile station in a digital mobile radio network and with the association tables containing the asssociation of individual bit vectors to be transmitted with space-time block symbols for use in a method according to one of claims 1 to
 8. 